Aerodynamic optimisation using gradient-based methods has found a wide range of academic applications in the last 30 years. This framework is also becoming more and more popular in the industrial world where, most of the time, unstructured grids are largely used. In this framework, apart from the need to solve the flow field, there is the need to quickly map the aerodynamic surface in terms of some aerodynamic figure of merits such as the drag coefficient, without being limited by the computational expense related to the grid size. This is a concrete industrial need which requires the efficient computation of the grid sensitivity.
A novel method based on the DGM (Delaunay Graph Mapping) mesh movement is proposed to efficiently compute the grid sensitivity required in the discrete adjoint optimisation framework. The method makes use of a one-to-one explicit algebraic mapping between the volume mesh and the solid boundary nodes. This procedure results in a straightforward computation of the gradient without the need to invert a large, sparse and stiff matrix generally associated with implicit mesh movements such as the spring or LE (Linear Elastic) analogy. The method is verified using FDs (Finite Difference) and a thorough comparison in terms of CPU time, formulation against the LE-based mesh movement and adjoint gradient is presented.
The DGM-based gradient chain allows to comfortably obtain the gradient with respect to each surface mesh point. Unfortunately, these gradients cannot be used directly because of their inherent poor smoothness feature. In order to address this issue one has to use a parameterisation technique which inevitably sacrifices the design space explorablity. To bridge the gap between the free-nodes and the parameterisation approaches, a novel formulation of the CST (Class Shape Transformation) was developed and termed l-CST (local-CST). The method is based on a simple trigonometric function which works as a cut-off filter on the BPs (Bernstein Polynomials) which are used to enforce a strong on-demand local control. The method is tested on an inverse geometric fitting and its effect on the resulting aerodynamic coefficients and the pressure distribution is also analysed.
The DGM-based chain allows the efficient mapping of the entire surface while the l-CST allows the combination of excellent explorablity and surface smoothness. The former is tested within the non-consistent mesh movement and sensitivity framework because there are situations where one method may be preferred over the other based on the grounds that mesh movement is a very different task than mesh sensitivity although strongly related to each other. The latter is instead tested against the free-nodes approach which offers a similar advantage in terms of discrete control although without maintaining a C2 curve unless properly smoothed.
